Answer: 2
Rank can be defined as:
$\equiv $ Linearly Independent Rows
$\equiv $ Linearly Independent Columns
$\equiv $ Pivot elements (count) in Echelon form of the Matrix
$\equiv $ Non-zero rows of Echelon form of the Matrix
Given matrix:
$\begin{bmatrix}
0 & 0 & -3\\
9 & 3 & 5\\
3 & 1& 1
\end{bmatrix}$
Let’s convert it to Echelon Form.
$\equiv \begin{bmatrix}
0 & 0 & -3\\
9 & 3 & 5\\
3 & 1& 1
\end{bmatrix} \xrightarrow{R_1 \leftrightarrow R_3} \begin{bmatrix}
3 & 1 & 1\\
9 & 3 & 5\\
0 & 0 & -3
\end{bmatrix}$
$\equiv \begin{bmatrix}
3 & 1 & 1\\
9 & 3 & 5\\
0 & 0 & -3
\end{bmatrix} \xrightarrow[]{R_2 \rightarrow R_2 - 3R_1} \begin{bmatrix}
3 & 1 & 1\\
0 & 0 & 2\\
0 & 0 & -3
\end{bmatrix}$
$\equiv \begin{bmatrix}
3 & 1 & 1\\
0 & 0 & 2\\
0 & 0 & -3
\end{bmatrix} \xrightarrow[]{R_3 → 3R_2 + 2R_3} \begin{bmatrix}
3 & 1 & 1\\
0 & 0 & 2\\
0 & 0 & 0
\end{bmatrix}$
In the resultant matrix, we have two Pivot elements as marked with $\color{Red} red$:
$\begin{bmatrix}
\color{Red} 3& 1 & 1\\
0 & 0 & \color{Red} 2\\
0 & 0 & 0
\end{bmatrix}$
Therefore, rank is 2.